1. P control [and first order processes] – part II
Control Theory
P control [and first order processes] – part II

2. Last week we ended here…
I want to design a control system for the process
using a P controller.
I choose the bias for a reference r=4 and a disturbance d=3: b=1.5.
I choose a control gain Kc=3.
What is the steady state error when r->5 and d=3?
What is the steady state error when r=4 and d->2?
u(t)
z(t) P
d(t)
r(t)

3. Group Task I want to design a control system for the process
using a P controller.
I choose the bias for a reference r=4 and a disturbance d=3: b=1.5.
I choose a control gain Kc=3.
Assume initially the desired equilibrium is reached.
z(t),u(t) for servo and control problem on previous slide?
u(t)
z(t) P
d(t)
r(t)

5. Result for servo problem (Kc=3)
z(t) t

6. Result for servo problem (Kc=3)
u(t) t

7. Group Task I want to design a control system for the process
using a P controller.
I choose the bias for a reference r=4 and a disturbance d=3: b=1.5.
I choose a control gain Kc=300.
Assume initially the desired equilibrium is reached.
z(t),u(t) for servo and control problem on slide 2?
u(t)
z(t) P
d(t)
r(t)

8. Result for servo problem (Kc=300)
z(t)
But… u(t) becomes already at t=1s !!!
Moreover, no real process is truely first order – see later…

9. One more time… Using P control, find the closed loop equationLC
Disturbance
Manipulated Variable
Using P control, find the closed loop equation
Elaborate on the size of the steady state error
for both servo and control problem!

10. P control if Kce+b < uA,min then uA=uA,min
MODEL OF THE P CONTROLLER:
if Kce+b < uA,min then uA=uA,min
if uA,min< Kce+b < uA,max then uA=Kce+b
if Kce+b > uA,max then uA=uA,max uA
[% tot. range]
100%
CONTROL CHARACTERISTIC
=> static behavior of the controller.
bias 0% r
 = r-ym [% tot. range]
r-100
prop
Definition: PROPORTIONAL BAND = 100/(Kc in %/%) = 100/Kc * (Δactuator)/(Δsensor)

11. Small exercise Kc = 3 mA/V ; b = 7 mA Sensor range: 0-10V
Actuator range: 4-20 mA
Proportional Band?

13. Some formulations of the definition
The proportional band is the error as percentage of range of the measured value to cause a full scale change in the controller output. [20sim]
Proportioning Band: is the area , in %, around the setpoint where the controller is actually controlling the process (regardless of the sensor range), i.e. the controller’s output is at some level other than 100% or 0%. [

14. Group Task 2 Process: We take A=1 m2 and c=0.1 m2/s Assume d=0.04 m3/s
Fin,u(t)
r(t)
Fout(t)
Fin,d(t)
We take A=1 m2 and c=0.1 m2/s
Assume d=0.04 m3/s
Actuator range:
4mA -> 0 m3/s
20mA -> 0.1 m3/s
Sensor range:
0.2m -> 0V
1.5m -> 5V
Draw the static process characteristic (x-axis: z in %, y axis: u in %)

15. Group Task 3 Process: Actuator: 4mA -> 0 m3/s 20mA -> 0.1 m3/s
Fin,u(t)
r(t)
Fout(t)
Fin,d(t)
Actuator:
4mA -> 0 m3/s
20mA -> 0.1 m3/s
Sensor:
0.2m -> 0V
1.5m -> 5V
Assume r = 1m
Choose the appropriate bias.
Next, draw the control characteristic for a PB of 75%.